Sturm-Liouville question

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In summary, the set {sin((pi)nx/a)} n=1 to infinity forms a basis for L2(0,a) in the Sturm-Liouville theory, as it satisfies the necessary conditions for a set of functions to form a basis and can be used to approximate any function in L2(0,a) with arbitrary precision. This is proven by showing that any polynomial function not equal to zero can be expressed as a linear combination of these basis functions, leading to the conclusion that the coefficients of the polynomial must be zero in order for the inner product to equal zero for all basis functions.
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I have a question that pertains to the Sturm-Liouville theory. Prove that {sin((pi)nx/a)} n=1 (a>0) is the basis for L2 (0,a).
 
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A basis set for L^2 (0,a)

I will assume we are dealing with continuous, real-valued functions here, as in your posted theorem. A set of functions, say [tex]\left\{ f_{n}(x)\right\}_{n=1}^{\infty},[/tex] forms a basis for L2(0,a) if and only if

for [tex]g(x)\in L^2(0,a)[/tex] we have [tex]\left< f_{n}(x),g(x)\right> =0[/tex] for every [tex]n=1,2,\ldots[/tex] implies that [tex]g(x)=0,[/tex]

where <,> denotes the inner product on [tex]L^2(0,a)[/tex] defined by [tex]\left< g(x) , h(x)\right> = \int_{0}^{a}g(x)h(x) dx[/tex]

In the above requirement for a set of functions to be basis, it should be understood that

[tex]\left< f_{n}(x),g(x)\right> =0[/tex] for every [tex]n=0,1,2,\ldots[/tex]

means that

[tex]\left< f_{0}(x),g(x)\right> =\left< f_{1}(x),g(x)\right> =\cdots =0[/tex]

or equivalently that g(x) is orthogonal to every fn(x),

the "implies that [tex]g(x)=0,[/tex]" part means that and the only function g(x) for which this condition may be satisfied (if the set of functions be a basis) is the zero function.

Now since we are dealing with continuous, real-valued functions, the Stone-Weierstrass theorem (in particular, Weierstrass approximation theorem) applies and any [tex]g(x)\in L^2(0,a)[/tex] can be approximated by a polynomial with arbitrary precision, hence we will test our set of functions against [tex]g(x)=c_0+c_1x+c_2x^2+\cdots +c_mx^m=\sum_{k=0}^{m}c_kx^k[/tex] for [tex]m=0,1,2,\ldots[/tex], where not every ck is zero. Here it goes:

[tex]\left< f_{n}(x) , g(x)\right> = \int_{0}^{a}\left( \sum_{k=0}^{m}c_kx^k \right) \sin\left( \frac{\pi nx}{a} \right) dx = \sum_{k=0}^{m}c_k\int_{0}^{a}x^k \sin\left( \frac{\pi nx}{a} \right) dx[/tex]

then prove that if the last integral is zero for every n=1,2,..., then [tex]c_k=0[/tex] for k=0,1,2,...m and you're done. You might try using [tex]\sin (x) = \frac{e^{ix}-e^{-ix}}{2i}[/tex] to evaluate the integral.
 

What is the Sturm-Liouville question?

The Sturm-Liouville question is a mathematical problem that involves finding the eigenfunctions and eigenvalues of a second-order linear differential equation. It is named after the mathematicians Jacques Charles François Sturm and Joseph Liouville, who independently worked on this problem in the 19th century.

What is the significance of the Sturm-Liouville question?

The Sturm-Liouville question has important applications in physics and engineering, particularly in solving boundary value problems. It also has connections to other areas of mathematics such as complex analysis and spectral theory.

What are the conditions for a Sturm-Liouville problem to have a unique solution?

In order for a Sturm-Liouville problem to have a unique solution, the differential equation must be self-adjoint, meaning that it is equal to its own adjoint. Additionally, the boundary conditions must be separated, meaning that they can be written as a sum of two functions that are only dependent on one of the variables.

What are the applications of the Sturm-Liouville question in physics?

The Sturm-Liouville question is used in physics to solve problems involving vibrational modes of a system, such as the motion of a vibrating string or the oscillations of a drumhead. It is also used in quantum mechanics to find the energy levels of a particle in a potential well.

What are some techniques for solving the Sturm-Liouville question?

Some common techniques for solving the Sturm-Liouville question include separation of variables, the method of Frobenius, and the Green's function method. Other techniques such as the Rayleigh-Ritz method and the variational method can also be used for more complex problems.

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